It's not oil. It is an excretion from the digestive track of scale insects. They could be on the bamboo, or on other nearby trees or shrubs, so you will need to verify which plant is infested in order to treat it. Use a systemic insecticide such as Bayer Advanced Tree and Shrub, this product is highly effective on scale insects. As far as the cleanup, any good detergent or soap should cut it. Try Dawn dish detergent or Murray's oil soap, both of those should clean without worrying about harming the dyes in the cushions. Spic and Span cleaner is great for hard surfaces but can fade/discolor some fabrics IMHO.

Staining isn't a property of the wood, it's a property of the finish on top of the wood. With any prefinished floor, unless you use some pretty harsh solvents, you're not going to alter the finish enough to allow actual stains to penetrate to the wood. However, any gloss finish over a very smooth wood will show any and all footprints, dirt, and other signs of everyday living. For your stated requirements, a natural white oak floor would be a better choice. It's inexpensive and has enough graining and is varied enough in coloration to hide dust and dog dirt. Bamboo would be too uniform in color and too smooth in texture to give equal hiding ability.

Why don't you post this on the paint forum and I am sure one of the experts there will have advice for you. I have a feeling you can stain bamboo and it would look wonderful since they sell bamboo floors in various stains.

Got a similar project in the works. Since bamboo is kinda sealed itself I plan to use Amber or 'Orange' Shellac made by Bull's Eye/ Zinsser. Get it at the depot. I've done lot's of work with this product and it's color is amazing after 5 or 6 coats.(the more the darker) It's very hard to work with on big surfaces as it sets up like laQuer and can get gummy. It won't look like mahogony but more like sunbaked varnish on teak. I've sprayed it as well on white painted walls and in the end after many coats I had 'Tortoise Shell' walls that were stunning. (done this for clients as well) Did my kitchen (doug fir) with it as well then varnished over for strength! Fun stuff. Do some experiments with throw away brushes. I also hear you can tint it! google that. Good luck.